Assume that \(G\cong PSL_{3}(3)\) . Then \(P\)  is maximal in \(G\)  by [1]},
and hence \(N_{G}(\left\langle \eta \right\rangle )=P\) . Then the
incidence structure \(\mathcal {D}=(\mathcal {P},\mathcal {B},\mathcal {I})\) 
above defined is flag-transitive with parameters \((v,b,k,r)=(144,468,12,39)\) 
by [2]}, Lemmas 1 and 2.
The group \(P\)  fixes exactly one element of \(\mathcal {P}\) , namely itself, and
\(\left\langle \eta \right\rangle \)  acts semiregularly on \(\mathcal {P-}\left\lbrace  P\right\rbrace  \) . It follows from [1]} that \(N_{G}(\left\langle \psi \right\rangle )\cong D_{18}\) . Hence, the number of elements in \(\mathcal {P}\) 
fixed by \(\left\langle \psi \right\rangle \)  is 6. Since \(P\)  is a Frobenius
group and since \(\left\langle \eta \right\rangle \)  acts semiregularly on \(\mathcal {P-}\left\lbrace  P\right\rbrace  \) , the five points fixed by \(\left\langle \psi \right\rangle \)  and distinct from \(P\)  lie in five distinct \(\left\langle \eta \right\rangle \) -orbits by [4]}, Proposition 4.2. Thus \(\mathcal {P}=\left\lbrace  Px:x\in G\right\rbrace  \)  is partitioned in one \(P\) -orbit of length 1,
five ones of length 13 and two ones of length 39.
Each \(P\) -orbit is of the form \(PyP\)  with \(y\in G\) . If we set \(\beta =\alpha ^{\gamma }\) , then \(\left\langle \alpha ,\beta \right\rangle \cong E_{4}\) ,
and it is not difficult to check that

\(P\alpha P\cap L=\left\lbrace  \alpha ,\gamma ^{\beta },\left( \gamma ^{\beta }\right) ^{-1}\right\rbrace , \; P\beta P\cap L=\left\lbrace  \beta ,\gamma ^{\alpha },\left( \gamma ^{\alpha }\right) ^{-1}\right\rbrace  \text{ and } \; PxP\cap L=\left\lbrace  x\right\rbrace  \) 
for each \(x\in \left\lbrace  1,\alpha \beta ,\gamma ,\gamma ^{-1},\gamma ^{\alpha \beta },\left( \gamma ^{\alpha \beta }\right) ^{-1}\right\rbrace  \) . Thus, the
double cosets \(PyP\)  with \(y\in \left\lbrace  1,\alpha ,\beta ,\alpha \beta ,\gamma ,\gamma ^{-1},\gamma ^{\alpha \beta },\left( \gamma ^{\alpha \beta }\right)^{-1}\right\rbrace  \)  are exactly all the eight \(P\) -orbits partitioning \(\mathcal {P}\) .
For each \(y\in \left\lbrace  \alpha ,\beta ,\alpha \beta ,\gamma ,\gamma ^{-1},\gamma ^{\alpha \beta },\left( \gamma ^{\alpha \beta }\right)^{-1}\right\rbrace  \) , the incidence structure \(\mathcal {K}_{y}=\left(PyP,LP\right) \)  is a tactical configuration by [5]}, 1.2.6. It follows from (REF )
that, the parameters \((v_{y},b_{y},k_{y},r_{y})\)  of \(\mathcal {K}_{y}\)  are either equal
to \((39,39,3,3)\) , or to \((13,39,1,3)\) , according to whether \(y\in \left\lbrace \alpha ,\beta \right\rbrace  \)  or \(y\in \left\lbrace  \alpha \beta ,\gamma ,\gamma ^{-1},\gamma ^{\alpha \beta },\left( \gamma ^{\alpha \beta }\right)^{-1}\right\rbrace  \)  respectively. Thus \(r_{y}=3\)  in each case,
and hence the number of blocks of \(\mathcal {D}\)  incident with \(P\)  and \(Px\) ,
where \(x\ne 1\) , is 3. Then 3 is the number of blocks incident with any two
distinct points of \(\mathcal {P}\) , as \(\mathcal {D}\)  is a flag-transitive
tactical configuration. Therefore, \(\mathcal {D}=(\mathcal {P},\mathcal {B})\) 
is a 2-\((12^{2},12,3)\)  design admitting \(G\)  as a flag-transitive
automorphism group.
Assume that \(\mathrm {Aut(\mathcal {D})}\ne G\) . Then \(G\trianglelefteq \mathrm {Aut(\mathcal {D})}\leqslant \mathrm {Aut(}G\mathrm {)}\) , where \(\mathrm {Aut(}G\mathrm {)}\cong G:\left\langle \sigma \right\rangle \)  and \(\sigma \)  is
defined above, as a consequence of the O'Nan-Scott Theorem (e.g. see [6]}, Theorem 4.1A), since \(v=2^{4}\cdot 3^{2}\) . Then \(\mathrm {Aut(\mathcal {D})}=\mathrm {Aut(}G\mathrm {)}\) , since \(G\ne \mathrm {Aut(\mathcal {D})}\)  by
our assumption. By [7]}, one \(\mathrm {Aut(}G\mathrm {)}_{P}\) -orbit on
\(\mathcal {P}\) , say \(\mathcal {O}\) , is of length 26 and is union of two \(G_{P}\) -orbits of length 13, the remaining \(\mathrm {Aut(}G\mathrm {)}_{P}\) -orbits on \(\mathcal {P}-\mathcal {O}\)  are also \(G_{P}\) -orbit. Thus, if \(\mathcal {B}_{P}\)  denotes the set of blocks of \(\mathcal {D}\)  incident with \(P\) , any block
in \(\mathcal {B}_{P}\)  intersects \(\mathcal {O}\)  in exactly 4 points. Hence, \((\mathcal {O},\mathcal {B}_{P})\)  is a tactical configuration by [5]}, 1.2.6, and its parameters are \((v_{0},b_{0},k_{0},r_{0})=(26,39,4,6)\) . So,
there are exactly 6 blocks of \(\mathcal {D}\)  incident with \(P\)  and with any
fixed point of \(\mathcal {O}\) . This is impossible, since \(\mathcal {D}\)  is a 2-design with \(\lambda =3\) . Thus \(\mathrm {Aut(\mathcal {D})}=G\) . Clearly, \(G\) 
is also the minimal flag-transitive automorphism group of \(\mathcal {D}\) , and
(1) is proven.
Finally, set \(A=\mathrm {Aut(}G\mathrm {)}\) . Then \(A\)  has a unique primitive
permutation representation of degree 144 and is the one on the set \(\mathcal {P}^{\prime }=\left\lbrace  N_{A}(P)x:x\in A\right\rbrace  \)  by [1]}. Also, \(G\)  permutes primitively the elements of \(\mathcal {P}^{\prime }\)  again by [1]}. Hence, we may consider the isomorphic copy \(\mathcal {D}_{0}=(\mathcal {P}^{\prime },\mathcal {B}_{0})\)  of the 2-design \(\mathcal {D}\)  constructed in
(1), where \(\mathcal {B}_{0}\)  consists of suitable 12-subsets of \(\mathcal {P}^{\prime }\) . Note that, \(\mathcal {B}_{0}^{\sigma }\ne \mathcal {B}_{0}\)  as \(\mathrm {Aut(\mathcal {D}}_{0}\mathrm {)}=G\)  by (1). Thus, \(\mathcal {D}_{0}^{\sigma }=(\mathcal {P}^{\prime },\mathcal {B}_{0}^{\sigma })\)  is a isomorphic to \(\mathcal {D}\) , but \(\mathcal {D}^{\sigma } \ne \mathcal {D}\) . Therefore, \(\mathcal {D}^{\prime }=(\mathcal {P}^{\prime },\mathcal {B}_{0}\cup \mathcal {B}_{0}^{\sigma })\)  is a 2-\((144,12,6)\)  design admitting \(A\)  as a
flag-transitive automorphism group. Arguing as in (1), we see that \(A\)  is
the unique flag-transitive automorphism group of \(\mathcal {D}^{\prime }\) .
If \(\mathcal {B}^{\prime }=\left\lbrace  Ly:y\in A\right\rbrace  \) , the incidence
structure \((\mathcal {P}^{\prime },\mathcal {B}^{\prime },\mathcal {I}^{\prime })\) , where
\(\mathcal {I}=\left\lbrace  (N_{A}(P)x,Ly)\in \mathcal {P}^{\prime }\times \mathcal {B}^{\prime }:N_{A}(P)x\cap Ly\ne \varnothing \right\rbrace  \text{,}\) 
is an isomorphic copy of \(\mathcal {D}^{\prime }\)  by [2]}, Lemmas 1, and
we obtain (2).
